2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/135516The purpose of the current paper is to introduce some new methods for studying the $p$-adic Banach spaces introduced by Emerton \cite{emerton}. We first relate these spaces to more familiar sheaf cohomology groups. As an application, we obtain a more general version of Emerton's spectral sequence. We also calculate the spaces in some easy cases. As a consequence, we obtain a number of vanishing theorems.30 pagesNumber Theory11F75; 11F33text